It's most like your 2nd example: it's an idiomatic expression meaning to get something to eat quickly, not generally a literal bite. Not necessarily a little bit to eat -- it could for example mean a sandwich for lunch -- but the idea is you don't have much time for eating or preparing. Your 3rd example doesn't make much sense, but in the context of your first one, you might say something like "would you like a bite?" (without "grab"), meaning perhaps a sample rather than a whole slice.

While it does use "bite", which would seem to be a small amount, it usually means to go out (to a restaurant) and get something to eat. It does not refer to specific food, and I usually hear it with "to eat" (as in "grab a bite to eat"). Your first example doesn't really sound natural since it's usually used to mean "go out to eat" and the third one is definitely wrong - the preposition "to" doesn't work there. The second example is fine, although I would expand it to "I'm so hungry; I'd like to grab a bite to eat." You generally hear this being used in a question, e.g. "Want to grab a bite to eat?"